Propagation of Singularities for Schrödinger Equations with Modestly Long Range Type Potentials

Abstract

In a previous paper by the second author [11], we discussed a characterization of the microlocal singularities for solutions to Schrödinger equations with long range type perturbations, using solutions to a Hamilton-Jacobi equation. In this paper we show that we may use Dollard type approximate solutions to the Hamilton-Jacobi equation if the perturbation satisfies somewhat stronger conditions. As applications, we describe the propagation of microlocal singularities for $e^{it{H}_0}{e}^{-itH}$ when the potential is asymptotically homogeneous as $|x|\to \infty$, where $H$ is our Schrödinger operator, and $H_0$ is the free Schrödinger operator, i.e., $H_0=-\frac{1}{2}△$. We show $e^{it{H}_0}{e}^{-itH}$ shifts the wave front set if the potential $V$ is asymptotically homogeneous of order 1, whereas $e^{itH}{e}^{-it{H}_0}$ is smoothing if $V$ is asymptotically homogenous of order $\beta \in (1,3/2)$.